A circle that contains the same tangent as that point and also the same curvature is known as an Osculating Circle. Become a Study. Try it risk-free for 30 days. Log in. Sign Up. Explore over 4, video courses. Find a degree that fits your goals.

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### Circle of Curvature

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Explore our homework questions and answers library Search. Browse Browse by subject. Ask a Question. To ask a site support question, click here.Radius of Curvature. The radius of curvature of a curve at a given point may be defined as the reciprocal of the curvature of the curve at that point.

Thus, if the radius of curvature is represented by Rthen. It should be noted that while the curvature K was defined as the absolute value of the respective fractions equivalent toK may be positive or negative.

Hence R may also be positive or negative, and will have the same sign as K. If R is positive, the curve is concave upwards at the particular point in question; if R is negative, the curve is concave downwards at that point. The radius of curvature may be thought of as the measure of the flatness or sharpness of a curve at a point; the smaller the radius of curvature, the sharper the curve.

The reader will see, from the following proof, why we take R equal to. Circle of Curvature. At a given point on a curve, there is, in general, but one tangent and one normal. An infinite number of circles, all having their centers lying on the normal, can be drawn through this point. Of these circles, the one whose radius equals the radius of curvature for the curve at that point is called the circle of curvature for the point.

Obviously, each point on the curve has a different circle of curvature except in special cases such as when the curve itself is a circle. The circle of curvature is also known as the osculating circle. In general, the circle of curvature of a curve at a point crosses the curve at that point.

The center of the circle of curvature is known as the center of curvature. Find the radius of curvature of the ellipse at the extremity of the minor axis, that is, at 0,2.

## Osculating circle

NOTE 2. NOTE 3. If we wish to determine the value of R at the extremity of the major axis, we should find that the value of at that point is infinite. In this case, we would interchange the axes, transforming the equation to and the extremity in question becomes 0,3.

See below, Exercise 8—2, Problem Find the radius of curvature for each of the following at the point indicated; in each case sketch the circle of curvature:. Compare the numerical values of the radii of curvature at the extremities of the two axes; what does this show about the comparative curvature at these two points? The Center of Curvature. Assume that the curve lies entirely on one side of the tangent.

Along the normal, toward the concave side of the curve, lay off the distance PQequal to the radius of curvature R at P. Thus Qthe center of the circle of curvature of the given curve for the point P, is called the center of curvature with respect to point P. Curvature Chapter Thus, if the radius of curvature is represented by Rthen Then we may at once write: It should be noted that while the curvature K was defined as the absolute value of the respective fractions equivalent toK may be positive or negative.

Therefore, 8— NOTE 1.The osculating circle of a curve at a given point is the circle that has the same tangent as at point as well as the same curvature. Just as the tangent line is the line best approximating a curve at a pointthe osculating circle is the best circle that approximates the curve at Grayp. Ignoring degenerate curves such as straight lines, the osculating circle of a given curve at a given point is unique.

Given a plane curve with parametric equations and parameterized by a variablethe radius of the osculating circle is simply the radius of curvature.

Here, derivatives are taken with respect to the parameter. In addition, let denote the circle passing through three points on a curve with.

Then the osculating circle is given by. Gardner, M. New York: W. Freeman, pp. Gray, A. Trott, M. The Mathematica GuideBook for Graphics. New York: Springer-Verlag, pp. Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own. Unlimited random practice problems and answers with built-in Step-by-step solutions.

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Osculating Circles. Circle of Curvature. Radius of Curvature of Catenary.In differential geometry of curvesthe osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p. Its center lies on the inner normal lineand its curvature is the same as that of the given curve at that point.

This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named circulus osculans Latin for "kissing circle" by Leibniz. The center and radius of the osculating circle at a given point are called center of curvature and radius of curvature of the curve at that point. A geometric construction was described by Isaac Newton in his Principia :. There being given, in any places, the velocity with which a body describes a given figure, by means of forces directed to some common centre: to find that centre.

**How to Find Osculating Circle**

Imagine a car moving along a curved road on a vast flat plane. Suddenly, at one point along the road, the steering wheel locks in its present position. Thereafter, the car moves in a circle that "kisses" the road at the point of locking. The curvature of the circle is equal to that of the road at that point. That circle is the osculating circle of the road curve at that point. This determines the unit tangent vector Tthe unit normal vector Nthe signed curvature k s and the radius of curvature at each point:.

The corresponding center of curvature is the point Q at distance R along Nin the same direction if k is positive and in the opposite direction if k is negative. If C is a regular space curve then the osculating circle is defined in a similar way, using the principal normal vector N. It lies in the osculating planethe plane spanned by the tangent and principal normal vectors T and N at the point P. Then the formulas for the signed curvature k tthe normal unit vector N tthe radius of curvature R tand the center Q t of the osculating circle are.

For a curve C given by a sufficiently smooth parametric equations twice continuously differentiablethe osculating circle may be obtained by a limiting procedure: it is the limit of the circles passing through three distinct points on C as these points approach P. The osculating circle S to a plane curve C at a regular point P can be characterized by the following properties:. This is usually expressed as "the curve and its osculating circle have the third or higher order contact" at P.

Loosely speaking, the vector functions representing C and S agree together with their first and second derivatives at P. If the derivative of the curvature with respect to s is nonzero at P then the osculating circle crosses the curve C at P. Points P at which the derivative of the curvature is zero are called vertices. If P is a vertex then C and its osculating circle have contact of order at least four.

If, moreover, the curvature has a non-zero local maximum or minimum at P then the osculating circle touches the curve C at P but does not cross it. The curve C may be obtained as the envelope of the one-parameter family of its osculating circles.

Their centers, i. Vertices of C correspond to singular points on its evolute. The parabola has fourth order contact with its osculating circle there. A Lissajous curve with ratio of frequencies can be parametrized as follows.

It has signed curvature k tnormal unit vector N t and radius of curvature R t given by. See the figure for an animation.Search this site.

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Now how can I solve the next part? Osculating circle approximates the curve in the neighborhood of a point, and you want the circle and the curve to have the same tangent vector and normal vector and, of course, the same curvature. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered.

Osculating circle Ask Question. Asked 3 years, 2 months ago. Active 3 years, 2 months ago. Viewed times. Matthew Conroy Hitman Hitman 2 2 silver badges 9 9 bronze badges. Perhaps you could describe the context in which this problem arose: is it from a book?

It never hurts to give more details: what book? Sharing details is always good. I looked at the pdf and I agree it is not clear what is being asked for. Perhaps a geometer and chime in. Active Oldest Votes. Andrew D. Hwang Andrew D. Hwang You have the center and radius of a circle, I think that's enough. It's a bit strange to know what curvature is and not to know what normal vector is. Don't get me wrong, I'm not judging you, but I'm interested to know how you defined curvature and the other things so that I can make this story a bit shorter.

And as far as I know vector field is different than a vector so I thought normal vector field is different than normal vector. I hope I've managed to explain why that's important. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Feedback post: New moderator reinstatement and appeal process revisions.In differential geometry of curvesthe osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p.

Its center lies on the inner normal lineand its curvature defines the curvature of the given curve at that point. This circle, which is the one among all tangent circles at the given point that approaches the curve most tightly, was named circulus osculans Latin for "kissing circle" by Leibniz. The center and radius of the osculating circle at a given point are called center of curvature and radius of curvature of the curve at that point. A geometric construction was described by Isaac Newton in his Principia :.

There being given, in any places, the velocity with which a body describes a given figure, by means of forces directed to some common centre: to find that centre. Imagine a car moving along a curved road on a vast flat plane.

Suddenly, at one point along the road, the steering wheel locks in its present position. Thereafter, the car moves in a circle that "kisses" the road at the point of locking. The curvature of the circle is equal to that of the road at that point. That circle is the osculating circle of the road curve at that point. This determines the unit tangent vector T sthe unit normal vector N sthe signed curvature k s and the radius of curvature R s at each point for which s is composed:.

The corresponding center of curvature is the point Q at distance R along Nin the same direction if k is positive and in the opposite direction if k is negative. If C is a regular space curve then the osculating circle is defined in a similar way, using the principal normal vector N. It lies in the osculating planethe plane spanned by the tangent and principal normal vectors T and N at the point P.

Then the formulas for the signed curvature k tthe normal unit vector N tthe radius of curvature R tand the center Q t of the osculating circle are.

If we do the calculations the results for the X and Y coordinates of the center of the osculating circle are:.

For a curve C given by a sufficiently smooth parametric equations twice continuously differentiablethe osculating circle may be obtained by a limiting procedure: it is the limit of the circles passing through three distinct points on C as these points approach P.

The osculating circle S to a plane curve C at a regular point P can be characterized by the following properties:. This is usually expressed as "the curve and its osculating circle have the second or higher order contact " at P.

Loosely speaking, the vector functions representing C and S agree together with their first and second derivatives at P. If the derivative of the curvature with respect to s is nonzero at P then the osculating circle crosses the curve C at P.

Points P at which the derivative of the curvature is zero are called vertices. If P is a vertex then C and its osculating circle have contact of order at least three. If, moreover, the curvature has a non-zero local maximum or minimum at P then the osculating circle touches the curve C at P but does not cross it.

The curve C may be obtained as the envelope of the one-parameter family of its osculating circles. Their centers, i. Vertices of C correspond to singular points on its evolute.

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